* Test problem 9.2.3 in the Test Collection Book
* Test problem 9.1.3 in the web page
* Test problem from Candler-Townsley 82

SET i /1*6/;
SET j /1*3/;

PARAMETER bigu;

bigu = 400;

VARIABLES

z, mu(j);

POSITIVE VARIABLES

x1, x2, x3, y1, y2, y3, y4, y5, y6, s(i), lb(i);

BINARY VARIABLES

yb(i);

EQUATIONS

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, kt1, kt2, kt3, kt4, kt5, kt6, cs1(i), cs2(i);

* Outer Objective function

c1..    4*y1 - 40*y2 - 4*y3 - 8*x1 - 4*x2 =e= z;

* Inner Problem Constraints

c2..  -y1 +   y2 +     y3 + y4                   =e= 1;

c3..  -y1 + 2*y2 - 0.5*y3      + y5      + 2*x1  =e= 1;

c4.. 2*y1 -   y2 - 0.5*y3           + y6 + 2*x2  =e= 1;

c5..  - y1 + s('1') =e= 0;

c6..  - y2 + s('2') =e= 0; 

c7..  - y3 + s('3') =e= 0;

c8..  - y4 + s('4') =e= 0;

c9..  - y5 + s('5') =e= 0;

c10.. - y6 + s('6') =e= 0; 

* KKT conditions for the inner problem optimum

kt1..  1 - mu('1') -     mu('2') +   2*mu('3') - lb('1') =e= 0;
kt2..  1 + mu('1') +   2*mu('2') -     mu('3') - lb('2') =e= 0;
kt3..  2 + mu('1') - 0.5*mu('2') - 0.5*mu('3') - lb('3') =e= 0;
kt4..  mu('1') - lb('4') =e= 0;
kt5..  mu('2') - lb('5') =e= 0;
kt6..  mu('3') - lb('6') =e= 0;

lb.up(i) = 200;
s.up(i)  = 200;
mu.up (j) = 200;
mu.lo(j) = -200;

* Complementarity Constraints

cs1(i).. lb(i) - bigu*yb(i) =l= 0;
cs2(i)..  s(i) - bigu*(1 - yb(i)) =l= 0;

MODEL BARDFALK/ALL/;

SOLVE BARDFALK USING MIP MINIMIZING z;